Theorem lubfun 18409

Description: The LUB is a function. (Contributed by NM, 9-Sep-2018.)

Hypothesis

Ref	Expression
lubfun.u	⊢ 𝑈 = (lub‘𝐾)

Assertion

Ref	Expression
lubfun	⊢ Fun 𝑈

Proof of Theorem lubfun

Dummy variables 𝑥 𝑠 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		funmpt 6578	. . . 4 ⊢ Fun (𝑠 ∈ 𝒫 (Base‘𝐾) ↦ (℩𝑥 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧))))
2		funres 6582	. . . 4 ⊢ (Fun (𝑠 ∈ 𝒫 (Base‘𝐾) ↦ (℩𝑥 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧)))) → Fun ((𝑠 ∈ 𝒫 (Base‘𝐾) ↦ (℩𝑥 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧)))) ↾ {𝑠 ∣ ∃!𝑥 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧))}))
3	1, 2	ax-mp 5	. . 3 ⊢ Fun ((𝑠 ∈ 𝒫 (Base‘𝐾) ↦ (℩𝑥 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧)))) ↾ {𝑠 ∣ ∃!𝑥 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧))})
4		eqid 2770	. . . . 5 ⊢ (Base‘𝐾) = (Base‘𝐾)
5		eqid 2770	. . . . 5 ⊢ (le‘𝐾) = (le‘𝐾)
6		lubfun.u	. . . . 5 ⊢ 𝑈 = (lub‘𝐾)
7		biid 264	. . . . 5 ⊢ ((∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧)) ↔ (∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧)))
8		id 23	. . . . 5 ⊢ (𝐾 ∈ V → 𝐾 ∈ V)
9	4, 5, 6, 7, 8	lubfval 18407	. . . 4 ⊢ (𝐾 ∈ V → 𝑈 = ((𝑠 ∈ 𝒫 (Base‘𝐾) ↦ (℩𝑥 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧)))) ↾ {𝑠 ∣ ∃!𝑥 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧))}))
10	9	funeqd 6562	. . 3 ⊢ (𝐾 ∈ V → (Fun 𝑈 ↔ Fun ((𝑠 ∈ 𝒫 (Base‘𝐾) ↦ (℩𝑥 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧)))) ↾ {𝑠 ∣ ∃!𝑥 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧))})))
11	3, 10	mpbiri 261	. 2 ⊢ (𝐾 ∈ V → Fun 𝑈)
12		fun0 6605	. . 3 ⊢ Fun ∅
13		fvprc 6877	. . . . 5 ⊢ (¬ 𝐾 ∈ V → (lub‘𝐾) = ∅)
14	6, 13	eqtrid 2817	. . . 4 ⊢ (¬ 𝐾 ∈ V → 𝑈 = ∅)
15	14	funeqd 6562	. . 3 ⊢ (¬ 𝐾 ∈ V → (Fun 𝑈 ↔ Fun ∅))
16	12, 15	mpbiri 261	. 2 ⊢ (¬ 𝐾 ∈ V → Fun 𝑈)
17	11, 16	pm2.61i 184	1 ⊢ Fun 𝑈

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   = wceq 1568   ∈ wcel 2150  {cab 2748  ∀wral 3086  ∃!wreu 3374  Vcvv 3462  ∅c0 4294  𝒫 cpw 4567   class class class wbr 5114   ↦ cmpt 5197   ↾ cres 5667  Fun wfun 6534  ‘cfv 6540  ℩crio 7370  Basecbs 17272  lecple 17320  lubclub 18368

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-9 2160  ax-10 2183  ax-11 2199  ax-12 2220  ax-ext 2742  ax-rep 5243  ax-sep 5262  ax-nul 5274  ax-pow 5340  ax-pr 5408

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2099  df-mo 2574  df-eu 2604  df-clab 2749  df-cleq 2762  df-clel 2845  df-nfc 2919  df-ne 2966  df-ral 3087  df-rex 3097  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3464  df-sbc 3753  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5560  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7371  df-lub 18403

This theorem is referenced by:  joinfval  18430  joinfval2  18431